Finite Group Databasebeta

← #29834326

Subgroups — #29834326

Order 15616.

748 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21456301C211yesmaximal, characteristic, complemented
4#21454666C211yesmaximal, characteristic, complemented
5#21447877C211yesmaximal, characteristic, complemented
6#21447876C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447855C211yesmaximal, characteristic, complemented
9#21447854C211yesmaximal, characteristic, complemented
10#21447853C211yesmaximal, characteristic, complemented
11#21447852C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic, complemented
14#21447525C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C2×#14519800C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840330—21no—
21#16840234—21no—
22#16840199—24no—
23C2×#14519720C2211yescharacteristic, complemented
24C2×D976C2211yescharacteristic
25C2×D976C2211yescharacteristic, complemented
26C2×#14519718C2211yescharacteristic
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic, complemented
29D16×D61—24no—
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840152—21no—
33#16840148—21no—
34#16840147C2211yescharacteristic, complemented
35#16840139C2211yescharacteristic, complemented
36#16840137C2211yescharacteristic, complemented
37#16839872—21no—
38#16839871—21no—
39C4×D488C2211yescharacteristic, complemented
40D8×Dic61C2211yescharacteristic, complemented
41#16839844C2211yescharacteristic, complemented
42#16839799C2211yescharacteristic
43#16839765—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839562C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×D4×D61—21no—
63C2×D4×D61—21no—
64#14519804—44no—
65#14519800—22no—
66#14519800—22no—
67C2×#11758201—21no—
68C2×#11758201—21no—
69C2×D488—21no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76D8×D61—22no—
77D8×D61—22no—
78D8×D61—22no—
79D8×D61D414yescomplemented
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519774—21no—
83#14519773—21no—
84#14519766—21no—
85#14519750—41no—
86#14519750—41no—
87#14519747—41no—
88#14519747—41no—
89C4×D244—21no—
90D4×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519735—21no—
95#14519734D411yescharacteristic
96C2×#11758198—41no—
97C22⋊C4×D61—41no—
98#14519720—22no—
99D976—22no—
100D976—22no—
101#14519718—22no—
102#14519718—22no—
103#14519718—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519708—21no—
110#14519699—21no—
111#14519698—21no—
112#14519697—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×SD16×C61—21no—
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120#124×C61—21no—
121C4×D4×C61—21no—
122SD32×C61—22no—
123D16×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#102×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C23×D61—41no—
131C2×Q8×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C22×C4×C61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C22×Dic61—41no—
140#11758226—42no—
141#11758226—42no—
142D4×D61—24no—
143D4×D61—42no—
144D4×D61—42no—
145D4×D61—42no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—41no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156Q16×C61—21no—
157Q16×C61—21no—
158SD16×C61—42no—
159D8×C61—21no—
160D8×C61—21no—
161D8×C61—21no—
162D8×C61C2×D412yes—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2411yescharacteristic
165C4⋊C4×C61—21no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C42×C61C2×D411yescharacteristic
170C22⋊Dic61—41no—
171#11758201—42no—
172#11758201—42no—
173#11758199—24no—
174#11758199—42no—
175#11758199—42no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198D811yescharacteristic
182#11758198D811yescharacteristic
183C4⋊Dic61—21no—
184#11758196D811yescharacteristic
185#11758196D811yescharacteristic
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190D488—21no—
191D488—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488—42no—
196D488C2×D412yes—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C23×C61—41no—
206C22×D61—41no—
207C22×D61—44no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—82no—
211C22×D61—161no—
212C22×D61—161no—
213C22×D61—161no—
214C22×D61—161no—
215C22×D61C2×D811yescharacteristic
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—161no—
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—44no—
229C22⋊D61—82no—
230C22⋊D61—161no—
231C22⋊D61—161no—
232C2×Dic61—41no—
233C2×Dic61—161no—
234C2×Dic61C2×D811yescharacteristic
235C2×Dic61C2×D811yescharacteristic
236C2×Dic61C2×D811yescharacteristic
237D244—22no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—82no—
244D244—161no—
245D244—161no—
246C4×D61—82no—
247C4×D61—161no—
248C4×D61—161no—
249C4×D61#12112yes—
250C4×D61AGL(1,ℤ/8)12yes—
251C8×C61—21no—
252C8×C61C22×D411yescharacteristic
253C8×C61C2×#4111yescharacteristic
254#92418—21no—
255#92418#12112yes—
256#33585—611nomaximal, Sylow
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61C22×D811yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—42no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—164no—
276C2×D61—164no—
277C2×D61#45912yes—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
284Dic61—21no—
285Dic61—21no—
286Dic61—81no—
287Dic61#45912yes—
288C2×#509—611no—
289C22×D16—611no—
290#3155—611no—
291#2948—611no—
292#2210—611no—
293#2210—611no—
294#2210—611no—
295#2210D6111yescharacteristic, complemented
296#2206—611no—
297#2206—611no—
298#2206—611no—
299#2206—611no—
300#2151—611no—
301#2140—611no—
302#2138—611no—
303C2×C61—82no—
304C2×C61—161no—
305C2×C61C2×#45911yescharacteristic
306C2×C61#341511yescharacteristic
307C2×C61#341511yescharacteristic
308D61—22no—
309D61—82no—
310D61—161no—
311D61—161no—
312C2×AGL(1,ℤ/8)—1221no—
313C2×#135—611no—
314C22×D8—611no—
315C22×D8—611no—
316#535—1221no—
317#518—1221no—
318#509—1224no—
319C2×SD32—611no—
320C2×SD32C2×D6111yescharacteristic, complemented
321C2×D16—611no—
322C2×D16—611no—
323C2×D16—611no—
324C2×D16—611no—
325C2×D16—611no—
326C2×D16—1224no—
327C2×D16C2×D6111yescharacteristic
328C2×M32—611no—
329C22×C16—611no—
330#496—1221no—
331#496—1221no—
332#495—611no—
333#495C2×D6111yescharacteristic, complemented
334#493—611no—
335#493—611no—
336#442—1221no—
337#442—1221no—
338C4×D8—611no—
339C4×D8—611no—
340C4×D8—611no—
341C4×D8C2×D6111yescharacteristic, complemented
342#427—611no—
343#418—1221no—
344#405—611no—
345C4⋊C16—611no—
346C4⋊C16—611no—
347C4⋊C16—611no—
348C4⋊C16C2×D6111yescharacteristic, complemented
349#358—611no—
350#358C2×D6111yescharacteristic, complemented
351#357—611no—
352#357—611no—
353#357—611no—
354#357—611no—
355#357—611no—
356#357C2×D6111yescharacteristic, complemented
357C61#3358511yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C22×D4—1221no—
360C22×D4—1221no—
361AGL(1,ℤ/8)—2444no—
362#135—1222no—
363#135—1222no—
364C2×Q16C22×D6111yescharacteristic
365C2×SD16—21no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×D8—614no—
369C2×D8—611no—
370C2×D8—611no—
371C2×D8—611no—
372C2×D8—1222no—
373C2×D8—1222no—
374C2×D8—1222no—
375C2×D8—1221no—
376C2×D8—1221no—
377C2×D8—1221no—
378C2×D8—1221no—
379C2×D8C22×D6111yescharacteristic
380C2×D8C22×D6111yescharacteristic
381C2×M16—611no—
382C22×C8—611no—
383#127—1221no—
384#127—1221no—
385#124—21no—
386#124—1221no—
387#121—2441no—
388#121—2441no—
389#120—2441no—
390#120—2441no—
391C4×D4—21no—
392C4×D4—1221no—
393C4×D4—1221no—
394C4×D4—1221no—
395C4×D4—2441no—
396C4×D4—2441no—
397#117—611no—
398#117—1221no—
399C2×C22⋊C4—2441no—
400C2×C22⋊C4—2441no—
401SD32—22no—
402SD32—1222no—
403D16—22no—
404D16—1222no—
405D16—1222no—
406D16—1222no—
407D16—1222no—
408D16—1222no—
409M32—1222no—
410C2×C16—611no—
411C2×C16—611no—
412C2×C16—1222no—
413C2×C16C22×D6111yescharacteristic
414C2×C16C22×D6111yescharacteristic
415#107—611no—
416#107C22×D6111yescharacteristic
417#106—1221no—
418#102—21no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422#97—611no—
423#97—611no—
424C4×C8—611no—
425C4×C8C22×D6111yescharacteristic
426C24—2441no—
427C24—2441no—
428#41—2442no—
429#41—2442no—
430C2×Q8—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—1224no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—2441no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C22×C4—41no—
448C22×C4—611no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453Q16—21no—
454Q16—21no—
455SD16—42no—
456SD16—2442no—
457SD16—2442no—
458D8—21no—
459D8—21no—
460D8—21no—
461D8—612no—
462D8—1224no—
463D8—1221no—
464D8—1221no—
465D8—1221no—
466D8—1221no—
467D8—1221no—
468D8—2442no—
469D8—2442no—
470D8—2442no—
471D8D4×D6112yes—
472M16—1222no—
473C2×C8—611no—
474C2×C8—612no—
475C2×C8—612no—
476C2×C8—611no—
477C2×C8D4×D6111yescharacteristic
478C2×C8C23×D6111yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—611no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—611no—
487C22⋊C4—611no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C42—611no—
495C42—2441no—
496C42D4×D6111yescharacteristic
497C16—21no—
498C16—21no—
499C16—1221no—
500C16—1221no—
501C23—41no—
502C23—41no—
503C23—611no—
504C23—9761no—
505C23—9761no—
506C23—9761no—
507C23—9761no—
508C23—4882no—
509C23—2444no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513Q8—41no—
514Q8—41no—
515D4—22no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—1222no—
520D4—9761no—
521D4—9761no—
522D4—9761no—
523D4—9761no—
524D4—4882no—
525D4—4882no—
526D4—2444no—
527D4—2441no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—161no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—9761no—
541C2×C4—9761no—
542C2×C4—9761no—
543C2×C4—4882no—
544C2×C4—2441no—
545C2×C4D8×D6111yescharacteristic
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—612no—
550C8—1221no—
551C8C2×D4×D6111yescharacteristic
552C8#41×D6111yescharacteristic
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—612no—
560C22—1221no—
561C22—1221no—
562C22—9764no—
563C22—9764no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22—2442no—
573C22C2×D8×D6111yescharacteristic
574C4—21no—
575C4—21no—
576C4—81no—
577C4—81no—
578C4—612no—
579C4—1221no—
580C4—1221no—
581C4—4881no—
582C4#121×D6111yescharacteristic
583C4AGL(1,ℤ/8)×D6111yescharacteristic
584C2—82no—
585C2—161no—
586C2—1222no—
587C2—9761no—
588C2—9761no—
589C2—4882no—
590C2#459×D6111yescharacteristic
591C2#2145895711yescharacteristic
592C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33585
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×D16×D61C2semidirect
3. #21456301C2semidirect
4. #21454666C2semidirect
5. #21447877C2semidirect
6. #21447876C2semidirect
7. #21447871C2semidirect
8. #21447855C2semidirect
9. #21447854C2semidirect
10. #21447853C2semidirect
11. #21447852C2semidirect
12. #21447576C2semidirect
13. #21447533C2semidirect
14. #21447525C2semidirect
15. C61×#2210C2semidirect
23. C2×#14519720C22semidirect
25. C2×D976C22semidirect
28. C2×#14519718C22semidirect
34. #16840147C22semidirect
35. #16840139C22semidirect
36. #16840137C22semidirect
39. C4×D488C22semidirect
40. D8×Dic61C22semidirect
41. #16839844C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839566C22semidirect
49. #16839564C22semidirect
50. #16839564C22semidirect
51. #16839563C22semidirect
52. #16839562C22semidirect
53. #16839562C22semidirect
54. C2×SD32×C61C22semidirect
56. C61×#495C22semidirect
57. C4×D8×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
79. D8×D61D4semidirect
295. #2210D61semidirect
320. C2×SD32C2×D61semidirect
333. #495C2×D61semidirect
341. C4×D8C2×D61semidirect
348. C4⋊C16C2×D61semidirect
350. #358C2×D61semidirect
356. #357C2×D61semidirect
357. C61#33585semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168401992
C2×D16×D612
#214563012
#214546662
#214478772
#214478762
#214478712
#214478552
#214478542
#214478532
#214478522
#214475762
#214475332
#214475252
C61×#22102
#3358561